Skip to content Skip to navigation

OpenStax-CNX

You are here: Home » Content » Digital-to-Digital Frequency Transformations

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Digital-to-Digital Frequency Transformations

Module by: Douglas L. Jones. E-mail the author

Given a prototype digital filter design, transformations similar to the bilinear transform can also be developed.

Requirements on such a mapping z-1=gz-1 z -1 g z -1 :

  1. points inside the unit circle stay inside the unit circle (condition to preserve stability)
  2. unit circle is mapped to itself (preserves frequency response)

This condition implies that e(i ω 1 )=ge(iω)=|gω|eigω ω 1 g ω g ω g ω requires that |ge(iω)|=1 g ω 1 on the unit circle!

Thus we require an all-pass transformation: gz-1= k =1pz-1 α k 1 α k z-1 g z -1 k 1 p z -1 α k 1 α k z -1 where | α K |<1 α K 1 , which is required to satisfy this condition.

Example 1: Lowpass-to-Lowpass

z 1 -1=z-1a1az-1 z 1 -1 z -1 a 1 a z -1 which maps original filter with a cutoff at ωc ωc to a new filter with cutoff ωc ωc , a=sin12( ω c ω c )sin12( ω c + ω c ) a 1 2 ω c ω c 1 2 ω c ω c

Example 2: Lowpass-to-Highpass

z 1 -1=z-1+a1+az-1 z 1 -1 z -1 a 1 a z -1 which maps original filter with a cutoff at ωc ωc to a frequency reversed filter with cutoff ωc ωc , a=cos12( ω c ω c )cos12( ω c + ω c ) a 1 2 ω c ω c 1 2 ω c ω c

(Interesting and occasionally useful!)

Content actions

Download module as:

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks